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Análise / Equações Diferenciais Parciais

 Título On fractional growth-dissipative Benjamin-Ono equations Expositor Ricardo Pastrán Universidad Nacional de Colombia, Bogotá Data Terça-feira, 11 de fevereiro de 2020, 15:30 Local Sala 232 Resumo

The form of some Benjamin-Ono type equations, which have been used in fluids and plasma theory, has motivated us to define, using fractional derivatives, a larger family of equations. This family of fractional growth-dissipative Benjamin-Ono equations is given by

$$u_t+\mathcal{H}u_{xx}-(\eta D_x^{\alpha}- \mu D_x^{\beta})u+uu_x=0,$$
where $u=u(x,t)$ is a real valued function, $0<\alpha < \beta$, $\eta \geq 0$, $\mu >0$, the operator $D_x^s$ is defined via the Fourier transform by $\widehat{D_x^s \varphi}(\xi)=|\xi|^s\,\widehat{\varphi}(\xi)$ and $\mathcal{H}$ denotes the Hilbert transform given by $\widehat{\mathcal{H}\varphi}(\xi)=-i \text{sgn} (\xi)\,\widehat{\varphi}(\xi)$, for all $\varphi \in \mathcal{S}(\mathbb{R})$.

For a wide class of parameters, taking into account dispersive and dissipative effects, we present well-posedness results in Sobolev spaces. For the cases $\alpha =1$, $\beta\geq 2$, and  $\eta=0$, $\beta\geq 2$, we show persistence properties of the solution flow in weighted Sobolev spaces $Z_{s,r}=H^s(\mathbb{R})\cap L^2(|x|^{2r}\,dx)$, for $s \geq r>0$, and some unique continuation properties in these spaces.

Joint work with Oscar Riaño (IMPA).